Installing the libraries:
library(tidyverse)
## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --
## <U+221A> ggplot2 3.3.2 <U+221A> purrr 0.3.4
## <U+221A> tibble 3.0.4 <U+221A> dplyr 1.0.2
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## <U+221A> readr 1.4.0 <U+221A> forcats 0.5.0
## -- Conflicts ------------------------------------------ tidyverse_conflicts() --
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library(tmap)
library(geojsonio)
##
## Attaching package: 'geojsonio'
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## pretty
library(plotly)
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## last_plot
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## layout
library(rgdal)
## Loading required package: sp
## rgdal: version: 1.5-18, (SVN revision 1082)
## Geospatial Data Abstraction Library extensions to R successfully loaded
## Loaded GDAL runtime: GDAL 3.0.4, released 2020/01/28
## Path to GDAL shared files: C:/Users/Elena/Documents/R/win-library/4.0/rgdal/gdal
## GDAL binary built with GEOS: TRUE
## Loaded PROJ runtime: Rel. 6.3.1, February 10th, 2020, [PJ_VERSION: 631]
## Path to PROJ shared files: C:/Users/Elena/Documents/R/win-library/4.0/rgdal/proj
## Linking to sp version:1.4-4
## To mute warnings of possible GDAL/OSR exportToProj4() degradation,
## use options("rgdal_show_exportToProj4_warnings"="none") before loading rgdal.
library(broom)
library(crosstalk)
library(sf)
## Linking to GEOS 3.8.0, GDAL 3.0.4, PROJ 6.3.1
library(sp)
library(spdep)
## Loading required package: spData
## To access larger datasets in this package, install the spDataLarge
## package with: `install.packages('spDataLarge',
## repos='https://nowosad.github.io/drat/', type='source')`
library(car)
## Loading required package: carData
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## Attaching package: 'car'
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library(fs)
library(janitor)
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## chisq.test, fisher.test
library(broom)
library(spgwr)
## NOTE: This package does not constitute approval of GWR
## as a method of spatial analysis; see example(gwr)
library(GWmodel)
## Loading required package: maptools
## Checking rgeos availability: TRUE
## Loading required package: robustbase
## Loading required package: Rcpp
## Loading required package: spatialreg
## Loading required package: Matrix
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## expand, pack, unpack
## Registered S3 methods overwritten by 'spatialreg':
## method from
## residuals.stsls spdep
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## summary.lagImpact spdep
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## print.sarlm spdep
## summary.sarlm spdep
## residuals.sarlm spdep
## deviance.sarlm spdep
## coef.sarlm spdep
## vcov.sarlm spdep
## fitted.sarlm spdep
## logLik.sarlm spdep
## anova.sarlm spdep
## predict.sarlm spdep
## print.summary.sarlm spdep
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## as.data.frame.sarlm.pred spdep
## residuals.spautolm spdep
## deviance.spautolm spdep
## coef.spautolm spdep
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## summary.spautolm spdep
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## create_WX, deviance.gmsar, deviance.sarlm, deviance.spautolm,
## deviance.stsls, do_ldet, eigen_pre_setup, eigen_setup, eigenw,
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## fitted.SFResult, fitted.spautolm, get.ClusterOption,
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## residuals.spautolm, residuals.stsls, sacsarlm, SE_classic_setup,
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## stsls, subgraph_eigenw, summary.gmsar, summary.sarlm,
## summary.spautolm, summary.stsls, trW, vcov.sarlm, Wald1.sarlm
## Welcome to GWmodel version 2.2-2.
## The new version of GWmodel 2.2-2 now is ready
library(spatialreg)
library(corrr)
library(corrplot)
## corrplot 0.84 loaded
source("http://www.sthda.com/upload/rquery_cormat.r")
2. Reading the OLS regression data:
Datareg <- readxl::read_xlsx('overall.xlsx')
## basic statistics
summary(Datareg[3:8])
## Percentage_of _Obese_Children_in_Year_6 Median_annual_income
## Min. :0.1226 Min. :23900
## 1st Qu.:0.2153 1st Qu.:27100
## Median :0.2352 Median :28200
## Mean :0.2267 Mean :30594
## 3rd Qu.:0.2503 3rd Qu.:32400
## Max. :0.3003 Max. :60000
## Population_per_square_km
## Min. : 2216
## 1st Qu.: 4577
## Median : 6049
## Mean : 7594
## 3rd Qu.:11167
## Max. :16038
## Percentage_of_homes_with_deficiency_in_access_to_nature
## Min. :0.0947
## 1st Qu.:0.1756
## Median :0.2593
## Mean :0.2783
## 3rd Qu.:0.3512
## Max. :1.0000
## Fast_food_outlets_per_100_residents
## Min. :0.06400
## 1st Qu.:0.07970
## Median :0.08960
## Mean :0.09446
## 3rd Qu.:0.10280
## Max. :0.17420
## Percentage_with_no_qualifications_aged_25_64
## Min. :0.00000
## 1st Qu.:0.04500
## Median :0.06100
## Mean :0.06148
## 3rd Qu.:0.07900
## Max. :0.11200
sd(Datareg$Percentage_with_no_qualifications_aged_25_64)
## [1] 0.02433377
sd(Datareg$`Percentage_of _Obese_Children_in_Year_6`)
## [1] 0.03772087
Merge boundaries and data:
LonBoroughs <- LondonBoroughs%>%
left_join(.,
Datareg,
by = c("GSS_CODE" = "Borough Code"))
Visualise the independent variables to observe their spatial variation:
tmap_mode("plot")
## tmap mode set to plotting
tm1 <- tm_shape(LonBoroughs) +
tm_polygons("Percentage_of _Obese_Children_in_Year_6",
style="jenks",
palette="Reds")+
tm_layout(main.title = "Chilhood Obesity in 2017-2018 per London Borough", main.title.position = "center", main.title.size = 1.4, legend.show=TRUE, legend.position=c("left","bottom"), legend.title.size = 0.8, legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata Β© Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm1
tmap_save(tm1, 'ChOb.png', dpi=300)
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\ChOb.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
tm2 <- tm_shape(LonBoroughs) +
tm_polygons("Median_annual_income",
style="jenks",
palette="Blues")+
tm_layout(main.title = "Personal Income by Tax Year 2017-2018", main.title.position = "center", main.title.size = 1.3, legend.show=TRUE, legend.position=c("left","bottom"), legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata Β© Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm2
tmap_save(tm2, 'Med_Income.png', dpi=300)
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\Med_Income.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
tm3 <- tm_shape(LonBoroughs) +
tm_polygons("Population_per_square_km",
style="quantile",
palette="YlOrRd")+
tm_layout(main.title = "Population Density in 2018 per London Borough", main.title.position = "center", main.title.size = 1.1, legend.show=TRUE, legend.position=c("left","bottom"), legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata Β© Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm3
tmap_save(tm3, 'Density.png', dpi=300)
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\Density.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
tm4 <- tm_shape(LonBoroughs) +
tm_polygons("Percentage_with_no_qualifications_aged_25_64",
style="quantile",
palette="Purples")+
tm_layout(main.title = "People aged 25-64 with no academic/professional qualifications in 2018", main.title.position = "center", main.title.size = 1.2, legend.show=TRUE, legend.position=c("left","bottom"), legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata Β© Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm4
tmap_save(tm4, 'Education.png', dpi=300)
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\Education.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
tm5 <- tm_shape(LonBoroughs) +
tm_polygons("Percentage_of_homes_with_deficiency_in_access_to_nature",
style="quantile",
palette="Greens")+
tm_layout(main.title = "Percentage of homes located within open space and access to nature in 2012", main.title.position = "center", main.title.size = 1.1, legend.show=TRUE, legend.position=c("left","bottom"),legend.title.size = 0.8,
legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata Β© Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm5
tmap_save(tm5, 'Nature.png', dpi=300)
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\Nature.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
tm6 <- tm_shape(LonBoroughs) +
tm_polygons("Fast_food_outlets_per_100_residents",
style="quantile",
palette="RdPu")+
tm_layout(main.title = "Density of fast food outlets at 02/07/2018", main.title.position = "center", main.title.size = 1.1, legend.show=TRUE, legend.position=c("left","bottom"),legend.title.size = 0.9,
legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata Β© Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm6
tmap_save(tm6, 'Fast_Food.png', dpi=300)
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\Fast_Food.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
Run regression for model 1:
model1 <- lm(LonBoroughs$`Percentage_of _Obese_Children_in_Year_6` ~
LonBoroughs$Median_annual_income +
LonBoroughs$Percentage_with_no_qualifications_aged_25_64 +
LonBoroughs$Population_per_square_km +
LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature+
LonBoroughs$Fast_food_outlets_per_100_residents)
summary(model1)
##
## Call:
## lm(formula = LonBoroughs$`Percentage_of _Obese_Children_in_Year_6` ~
## LonBoroughs$Median_annual_income + LonBoroughs$Percentage_with_no_qualifications_aged_25_64 +
## LonBoroughs$Population_per_square_km + LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature +
## LonBoroughs$Fast_food_outlets_per_100_residents)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.054430 -0.019171 0.001383 0.015530 0.053451
##
## Coefficients:
## Estimate
## (Intercept) 1.816e-01
## LonBoroughs$Median_annual_income -1.944e-06
## LonBoroughs$Percentage_with_no_qualifications_aged_25_64 4.980e-01
## LonBoroughs$Population_per_square_km 2.054e-06
## LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature 8.959e-02
## LonBoroughs$Fast_food_outlets_per_100_residents 3.534e-01
## Std. Error
## (Intercept) 4.510e-02
## LonBoroughs$Median_annual_income 1.307e-06
## LonBoroughs$Percentage_with_no_qualifications_aged_25_64 3.109e-01
## LonBoroughs$Population_per_square_km 1.539e-06
## LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature 4.243e-02
## LonBoroughs$Fast_food_outlets_per_100_residents 2.940e-01
## t value
## (Intercept) 4.027
## LonBoroughs$Median_annual_income -1.488
## LonBoroughs$Percentage_with_no_qualifications_aged_25_64 1.602
## LonBoroughs$Population_per_square_km 1.334
## LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature 2.112
## LonBoroughs$Fast_food_outlets_per_100_residents 1.202
## Pr(>|t|)
## (Intercept) 0.000412
## LonBoroughs$Median_annual_income 0.148405
## LonBoroughs$Percentage_with_no_qualifications_aged_25_64 0.120770
## LonBoroughs$Population_per_square_km 0.193285
## LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature 0.044110
## LonBoroughs$Fast_food_outlets_per_100_residents 0.239683
##
## (Intercept) ***
## LonBoroughs$Median_annual_income
## LonBoroughs$Percentage_with_no_qualifications_aged_25_64
## LonBoroughs$Population_per_square_km
## LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature *
## LonBoroughs$Fast_food_outlets_per_100_residents
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02865 on 27 degrees of freedom
## Multiple R-squared: 0.5131, Adjusted R-squared: 0.423
## F-statistic: 5.691 on 5 and 27 DF, p-value: 0.001039
tidy(model1)
## # A tibble: 6 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 1.82e-1 4.51e-2 4.03 4.12e-4
## 2 LonBoroughs$Median_annual_income -1.94e-6 1.31e-6 -1.49 1.48e-1
## 3 LonBoroughs$Percentage_with_no_qualifi~ 4.98e-1 3.11e-1 1.60 1.21e-1
## 4 LonBoroughs$Population_per_square_km 2.05e-6 1.54e-6 1.33 1.93e-1
## 5 LonBoroughs$Percentage_of_homes_with_d~ 8.96e-2 4.24e-2 2.11 4.41e-2
## 6 LonBoroughs$Fast_food_outlets_per_100_~ 3.53e-1 2.94e-1 1.20 2.40e-1
glance(model1)
## # A tibble: 1 x 12
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.513 0.423 0.0287 5.69 0.00104 5 73.7 -133. -123.
## # ... with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
Histograms and transformations
We’ll plot some histograms to check if the variables follow normal distribution:
hist(LonBoroughs$`Percentage_of _Obese_Children_in_Year_6`,
main="Figure 2: A histogram of the distribution of the Childhood Obesity variable",
xlab="% of Obese Children in Year 6",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughs$`Percentage_of _Obese_Children_in_Year_6`), col="red", lwd=3)
hist(LonBoroughs$Median_annual_income,
main="Figure 3: A histogram of median annual income variable",
xlab="median annual income",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughs$Median_annual_income), col="red", lwd=3)
hist(LonBoroughs$Population_per_square_km,
main="Figure 6: A histogram of population density variable",
xlab="population per square km",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughs$Population_per_square_km), col="red", lwd=3)
hist(LonBoroughs$Percentage_with_no_qualifications_aged_25_64,
main="Figure 9: A histogram of people without qualifications variable",
xlab="% of people with no qualifications aged 25-64",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughs$Percentage_with_no_qualifications_aged_25_64), col="red", lwd=3)
hist(LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature,
main="Figure 10: A histogram of access to nature variable",
xlab="% of home with deficient access to nature",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature), col="red", lwd=3)
hist(LonBoroughs$Fast_food_outlets_per_100_residents,
main="Figure 13: A histogram of density of Fast Food outlets variable",
xlab="Fast Food outlets per 100 residents",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughs$Fast_food_outlets_per_100_residents), col="red", lwd=3)
Plot histograms of transformed variables:
hist(LonBoroughstrans$`Median_annual_income _^-2.5`,
main="Figure 5: A histogram of (median annual income)^-2.5 variable",
xlab="(median annual income)^-2.5",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughstrans$`Median_annual_income _^-2.5`), col="red", lwd=3)
hist(LonBoroughstrans$log_Population_per_square_km,
main="Figure 8: A histogram of log population density variable",
xlab="log(population per square km)",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughstrans$log_Population_per_square_km), col="red", lwd=3)
hist((LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature)^-0.5,
main="Figure 12: A histogram of (access to nature)^-0.5 variable",
xlab="(% of home with deficient access to nature)^-0.5",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density((LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature)^-0.5), col="red", lwd=3)
hist(LonBoroughstrans$`Fast_food_outlets_per_100_residents_^-1`,
main="Figure 15: A histogram of density of (Fast Food outlets)^-1 variable",
xlab="(Fast Food outlets per 100 residents)^-1",
border="black",
col="white",
las=1,
breaks=20,
freq = FALSE)
lines(density(LonBoroughstrans$`Fast_food_outlets_per_100_residents_^-1`), col="red", lwd=3)
Run regression for model 2:
model2 <- lm(Data_reg_trans$`Percentage_of _Obese_Children_in_Year_6` ~
Data_reg_trans$`Median_annual_income _^-2.5` +
Data_reg_trans$Percentage_with_no_qualifications_aged_25_64 +
Data_reg_trans$log_Population_per_square_km +
Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature +
Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`)
summary(model2)
##
## Call:
## lm(formula = Data_reg_trans$`Percentage_of _Obese_Children_in_Year_6` ~
## Data_reg_trans$`Median_annual_income _^-2.5` + Data_reg_trans$Percentage_with_no_qualifications_aged_25_64 +
## Data_reg_trans$log_Population_per_square_km + Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature +
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.047962 -0.011960 -0.000369 0.007922 0.045922
##
## Coefficients:
## Estimate
## (Intercept) -2.503e-02
## Data_reg_trans$`Median_annual_income _^-2.5` 8.969e+09
## Data_reg_trans$Percentage_with_no_qualifications_aged_25_64 2.238e-01
## Data_reg_trans$log_Population_per_square_km 5.375e-02
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 7.519e-02
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` -4.507e-03
## Std. Error
## (Intercept) 9.309e-02
## Data_reg_trans$`Median_annual_income _^-2.5` 2.285e+09
## Data_reg_trans$Percentage_with_no_qualifications_aged_25_64 2.214e-01
## Data_reg_trans$log_Population_per_square_km 1.963e-02
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 3.007e-02
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` 2.249e-03
## t value
## (Intercept) -0.269
## Data_reg_trans$`Median_annual_income _^-2.5` 3.926
## Data_reg_trans$Percentage_with_no_qualifications_aged_25_64 1.011
## Data_reg_trans$log_Population_per_square_km 2.738
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 2.501
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` -2.004
## Pr(>|t|)
## (Intercept) 0.790093
## Data_reg_trans$`Median_annual_income _^-2.5` 0.000538
## Data_reg_trans$Percentage_with_no_qualifications_aged_25_64 0.321110
## Data_reg_trans$log_Population_per_square_km 0.010815
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 0.018761
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` 0.055202
##
## (Intercept)
## Data_reg_trans$`Median_annual_income _^-2.5` ***
## Data_reg_trans$Percentage_with_no_qualifications_aged_25_64
## Data_reg_trans$log_Population_per_square_km *
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature *
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02271 on 27 degrees of freedom
## Multiple R-squared: 0.694, Adjusted R-squared: 0.6374
## F-statistic: 12.25 on 5 and 27 DF, p-value: 2.935e-06
tidy(model2)
## # A tibble: 6 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -2.50e-2 9.31e-2 -0.269 7.90e-1
## 2 Data_reg_trans$`Median_annual_income _^-~ 8.97e+9 2.28e+9 3.93 5.38e-4
## 3 Data_reg_trans$Percentage_with_no_qualif~ 2.24e-1 2.21e-1 1.01 3.21e-1
## 4 Data_reg_trans$log_Population_per_square~ 5.37e-2 1.96e-2 2.74 1.08e-2
## 5 Data_reg_trans$Percentage_of_homes_with_~ 7.52e-2 3.01e-2 2.50 1.88e-2
## 6 Data_reg_trans$`Fast_food_outlets_per_10~ -4.51e-3 2.25e-3 -2.00 5.52e-2
glance(model2)
## # A tibble: 1 x 12
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.694 0.637 0.0227 12.2 2.94e-6 5 81.4 -149. -138.
## # ... with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
We’ll exclude the statistically insignificant variable and re-run the regression.
Run regression for model 3:
model3 <- lm(Data_reg_trans$`Percentage_of _Obese_Children_in_Year_6` ~
Data_reg_trans$`Median_annual_income _^-2.5` +
Data_reg_trans$log_Population_per_square_km +
Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature +
Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`)
summary(model3)
##
## Call:
## lm(formula = Data_reg_trans$`Percentage_of _Obese_Children_in_Year_6` ~
## Data_reg_trans$`Median_annual_income _^-2.5` + Data_reg_trans$log_Population_per_square_km +
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature +
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.050502 -0.012735 0.000638 0.011500 0.050048
##
## Coefficients:
## Estimate
## (Intercept) -3.911e-02
## Data_reg_trans$`Median_annual_income _^-2.5` 1.044e+10
## Data_reg_trans$log_Population_per_square_km 5.943e-02
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 7.544e-02
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` -4.873e-03
## Std. Error
## (Intercept) 9.208e-02
## Data_reg_trans$`Median_annual_income _^-2.5` 1.762e+09
## Data_reg_trans$log_Population_per_square_km 1.882e-02
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 3.008e-02
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` 2.220e-03
## t value
## (Intercept) -0.425
## Data_reg_trans$`Median_annual_income _^-2.5` 5.925
## Data_reg_trans$log_Population_per_square_km 3.158
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 2.508
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` -2.195
## Pr(>|t|)
## (Intercept) 0.67426
## Data_reg_trans$`Median_annual_income _^-2.5` 2.24e-06
## Data_reg_trans$log_Population_per_square_km 0.00379
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature 0.01820
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` 0.03662
##
## (Intercept)
## Data_reg_trans$`Median_annual_income _^-2.5` ***
## Data_reg_trans$log_Population_per_square_km **
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature *
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1` *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.02272 on 28 degrees of freedom
## Multiple R-squared: 0.6825, Adjusted R-squared: 0.6371
## F-statistic: 15.04 on 4 and 28 DF, p-value: 1.118e-06
tidy(model3)
## # A tibble: 5 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -3.91e- 2 9.21e-2 -0.425 6.74e-1
## 2 Data_reg_trans$`Median_annual_income _~ 1.04e+10 1.76e+9 5.92 2.24e-6
## 3 Data_reg_trans$log_Population_per_squa~ 5.94e- 2 1.88e-2 3.16 3.79e-3
## 4 Data_reg_trans$Percentage_of_homes_wit~ 7.54e- 2 3.01e-2 2.51 1.82e-2
## 5 Data_reg_trans$`Fast_food_outlets_per_~ -4.87e- 3 2.22e-3 -2.20 3.66e-2
glance(model3)
## # A tibble: 1 x 12
## r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.682 0.637 0.0227 15.0 1.12e-6 4 80.8 -150. -141.
## # ... with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
Regression Assumptions for model 3:
a. Plot regression fit line
q <- qplot(x = `Median_annual_income _^-2.5`,
y = `Percentage_of _Obese_Children_in_Year_6`,
data=LonBoroughstrans)
q + stat_smooth(method="lm", se=FALSE, size=1) +
geom_jitter()
## `geom_smooth()` using formula 'y ~ x'
q1 <- qplot(x = log_Population_per_square_km,
y = `Percentage_of _Obese_Children_in_Year_6`,
data=LonBoroughstrans)
q1 + stat_smooth(method="lm", se=FALSE, size=1) +
geom_jitter()
## `geom_smooth()` using formula 'y ~ x'
q2 <- qplot(x = Percentage_of_homes_with_deficiency_in_access_to_nature,
y = `Percentage_of _Obese_Children_in_Year_6`,
data=LonBoroughstrans)
q2 + stat_smooth(method="lm", se=FALSE, size=1) +
geom_jitter()
## `geom_smooth()` using formula 'y ~ x'
q3 <- qplot(x = `Fast_food_outlets_per_100_residents_^-1`,
y = `Percentage_of _Obese_Children_in_Year_6`,
data=LonBoroughstrans)
q3 + stat_smooth(method="lm", se=FALSE, size=1) +
geom_jitter()
## `geom_smooth()` using formula 'y ~ x'
b. Check the residuals into our dataframe
# Save the residuals into our dataframe
model_data <- model3 %>%
augment(., Data_reg_trans)
# Add them to the shapelayer
LonBoroughstrans <- LonBoroughstrans %>%
mutate(model3resids = residuals(model3))
# Plot residuals to see if they follow normal distribution
model_data%>%
dplyr::select(.resid)%>%
pull()%>%
qplot()+
geom_histogram(bins=30)
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
c. Visualisation of the correlation matrix
# library corrplot + rquery.cormat
rquery.cormat(Data_reg_trans[4:7], type='full',
graph=TRUE, graphType="correlogram")
## $r
## Median_annual_income _^-2.5
## Median_annual_income _^-2.5 1.00
## Fast_food_outlets_per_100_residents_^-1 0.23
## log_Population_per_square_km -0.11
## Percentage_of_homes_with_deficiency_in_access_to_nature -0.21
## Fast_food_outlets_per_100_residents_^-1
## Median_annual_income _^-2.5 0.23
## Fast_food_outlets_per_100_residents_^-1 1.00
## log_Population_per_square_km -0.35
## Percentage_of_homes_with_deficiency_in_access_to_nature -0.47
## log_Population_per_square_km
## Median_annual_income _^-2.5 -0.11
## Fast_food_outlets_per_100_residents_^-1 -0.35
## log_Population_per_square_km 1.00
## Percentage_of_homes_with_deficiency_in_access_to_nature -0.16
## Percentage_of_homes_with_deficiency_in_access_to_nature
## Median_annual_income _^-2.5 -0.21
## Fast_food_outlets_per_100_residents_^-1 -0.47
## log_Population_per_square_km -0.16
## Percentage_of_homes_with_deficiency_in_access_to_nature 1.00
##
## $p
## Median_annual_income _^-2.5
## Median_annual_income _^-2.5 0.00
## Fast_food_outlets_per_100_residents_^-1 0.20
## log_Population_per_square_km 0.55
## Percentage_of_homes_with_deficiency_in_access_to_nature 0.25
## Fast_food_outlets_per_100_residents_^-1
## Median_annual_income _^-2.5 0.2000
## Fast_food_outlets_per_100_residents_^-1 0.0000
## log_Population_per_square_km 0.0490
## Percentage_of_homes_with_deficiency_in_access_to_nature 0.0059
## log_Population_per_square_km
## Median_annual_income _^-2.5 0.550
## Fast_food_outlets_per_100_residents_^-1 0.049
## log_Population_per_square_km 0.000
## Percentage_of_homes_with_deficiency_in_access_to_nature 0.380
## Percentage_of_homes_with_deficiency_in_access_to_nature
## Median_annual_income _^-2.5 0.2500
## Fast_food_outlets_per_100_residents_^-1 0.0059
## log_Population_per_square_km 0.3800
## Percentage_of_homes_with_deficiency_in_access_to_nature 0.0000
##
## $sym
## Median_annual_income _^-2.5
## Median_annual_income _^-2.5 1
## Fast_food_outlets_per_100_residents_^-1
## log_Population_per_square_km
## Percentage_of_homes_with_deficiency_in_access_to_nature
## Fast_food_outlets_per_100_residents_^-1
## Median_annual_income _^-2.5
## Fast_food_outlets_per_100_residents_^-1 1
## log_Population_per_square_km .
## Percentage_of_homes_with_deficiency_in_access_to_nature .
## log_Population_per_square_km
## Median_annual_income _^-2.5
## Fast_food_outlets_per_100_residents_^-1
## log_Population_per_square_km 1
## Percentage_of_homes_with_deficiency_in_access_to_nature
## Percentage_of_homes_with_deficiency_in_access_to_nature
## Median_annual_income _^-2.5
## Fast_food_outlets_per_100_residents_^-1
## log_Population_per_square_km
## Percentage_of_homes_with_deficiency_in_access_to_nature 1
## attr(,"legend")
## [1] 0 ' ' 0.3 '.' 0.6 ',' 0.8 '+' 0.9 '*' 0.95 'B' 1
vif(model1)
## LonBoroughs$Median_annual_income
## 3.027936
## LonBoroughs$Percentage_with_no_qualifications_aged_25_64
## 2.230273
## LonBoroughs$Population_per_square_km
## 1.537062
## LonBoroughs$Percentage_of_homes_with_deficiency_in_access_to_nature
## 1.923653
## LonBoroughs$Fast_food_outlets_per_100_residents
## 1.911991
vif(model2)
## Data_reg_trans$`Median_annual_income _^-2.5`
## 1.811432
## Data_reg_trans$Percentage_with_no_qualifications_aged_25_64
## 1.800544
## Data_reg_trans$log_Population_per_square_km
## 1.465018
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature
## 1.537162
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`
## 1.731994
vif(model3)
## Data_reg_trans$`Median_annual_income _^-2.5`
## 1.076685
## Data_reg_trans$log_Population_per_square_km
## 1.344984
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature
## 1.537056
## Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`
## 1.686878
d. Print some model diagnositcs for homoscedasticity
par(mfrow=c(2,2))
#plot to 2 by 2 array
plot(model3)
e.Independence of errors
# Standard autocorrelation: run durbin-watson test
DW <- durbinWatsonTest(model3)
tidy(DW)
## # A tibble: 1 x 5
## statistic p.value autocorrelation method alternative
## <dbl> <dbl> <dbl> <chr> <chr>
## 1 2.19 0.644 -0.140 Durbin-Watson Test two.sided
# Plot the residuals to check spatial autocorrelation
tmap_mode("plot")
## tmap mode set to plotting
tm_res <- tm_shape(LonBoroughstrans) +
tm_polygons("model3resids", palette = "RdYlBu")+
tm_layout(main.title = "Map of residuals of Model 3", main.title.position = "center", main.title.size = 1.4,
legend.show=TRUE, legend.position=c("left","bottom"), legend.title.size = 0.9, legend.frame=TRUE)+
tm_scale_bar(position=c("right", "top"), text.size=0.6)+
tm_compass(north=0,type="8star", position=c("left", "top"))+
tm_credits("Contains National Statistics\nand Ordnance Survey\ndata © Crown copyright\nand database right [2015]\nDepartment: CASA-UCL\nCourse:GIS, Professors:\nA.Denett & A.Maclachlan\nCreated by: Eleni Kalantzi\nDate: 27/12/20", position=c("right", "bottom"))
tm_res
## Variable(s) "model3resids" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
tmap_save(tm_res, 'Static_resid_3.png', dpi=300)
## Variable(s) "model3resids" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
## Map saved to C:\Users\Elena\OneDrive - University College London\CASA_2020\modules\GIS\final_coursework\Reg\GIS_Coursework\Static_resid_3.png
## Resolution: 2389.896 by 1845.268 pixels
## Size: 7.966321 by 6.150895 inches (300 dpi)
Start calculating Moran’s I:
# Calculate the centroids of all Boroughs in London
coordsB <- LondonBoroughs%>%
st_centroid()%>%
st_geometry()
## Warning in st_centroid.sf(.): st_centroid assumes attributes are constant over
## geometries of x
plot(coordsB,axes=TRUE)
# create a neighbours list
LB_nb <- LonBoroughstrans %>%
poly2nb(., queen=T)
# and 4 nearest neighbours
knn_boroughs <-coordsB %>%
knearneigh(., k=4)
LB_knn <- knn_boroughs %>%
knn2nb()
# then plot them
plot(LB_nb, st_geometry(coordsB), col="red")
plot(LondonBoroughs$geometry, add=T)
plot(LB_knn, st_geometry(coordsB), col="blue")
plot(LondonBoroughs$geometry, add=T)
# Create a spatial weights matrix object from these weights
LB.queens_weight <- LB_nb %>%
nb2listw(., style="C")
LB.knn_4_weight <- LB_knn %>%
nb2listw(., style="C")
LB.knn_4_weight
## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 33
## Number of nonzero links: 132
## Percentage nonzero weights: 12.12121
## Average number of links: 4
## Non-symmetric neighbours list
##
## Weights style: C
## Weights constants summary:
## n nn S0 S1 S2
## C 33 1089 33 14.25 138
LB.queens_weight
## Characteristics of weights list object:
## Neighbour list object:
## Number of regions: 33
## Number of nonzero links: 136
## Percentage nonzero weights: 12.48852
## Average number of links: 4.121212
##
## Weights style: C
## Weights constants summary:
## n nn S0 S1 S2
## C 33 1089 33 16.01471 143.6613
# Moran's I for Queen and Knn4 for residuals
Queen <- LonBoroughstrans %>%
st_drop_geometry()%>%
dplyr::select(model3resids)%>%
pull()%>%
moran.test(., LB.queens_weight)%>%
tidy()
Nearest_neighbour <- LonBoroughstrans %>%
st_drop_geometry()%>%
dplyr::select(model3resids)%>%
pull()%>%
moran.test(., LB.knn_4_weight)%>%
tidy()
Queen
## # A tibble: 1 x 7
## estimate1 estimate2 estimate3 statistic p.value method alternative
## <dbl> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 -0.0810 -0.0312 0.0123 -0.448 0.673 Moran I test unde~ greater
Nearest_neighbour
## # A tibble: 1 x 7
## estimate1 estimate2 estimate3 statistic p.value method alternative
## <dbl> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 -0.0691 -0.0312 0.0109 -0.362 0.641 Moran I test unde~ greater
# Moran's I for model 3
lm.morantest(model3, LB.knn_4_weight)
##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = Data_reg_trans$`Percentage_of
## _Obese_Children_in_Year_6` ~ Data_reg_trans$`Median_annual_income
## _^-2.5` + Data_reg_trans$log_Population_per_square_km +
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature
## + Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`)
## weights: LB.knn_4_weight
##
## Moran I statistic standard deviate = -0.37084, p-value = 0.6446
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## -0.06905526 -0.02996277 0.01111244
lm.morantest(model3, LB.queens_weight)
##
## Global Moran I for regression residuals
##
## data:
## model: lm(formula = Data_reg_trans$`Percentage_of
## _Obese_Children_in_Year_6` ~ Data_reg_trans$`Median_annual_income
## _^-2.5` + Data_reg_trans$log_Population_per_square_km +
## Data_reg_trans$Percentage_of_homes_with_deficiency_in_access_to_nature
## + Data_reg_trans$`Fast_food_outlets_per_100_residents_^-1`)
## weights: LB.queens_weight
##
## Moran I statistic standard deviate = -0.40157, p-value = 0.656
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I Expectation Variance
## -0.08102434 -0.03573773 0.01271824
3. Spatial Regression models
We’ll run regression for Spatial lag model first:
# Spatial lag model for queen
slag_dv_model3_queen <- lagsarlm(`Percentage_of _Obese_Children_in_Year_6` ~ `Median_annual_income _^-2.5` +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
`Fast_food_outlets_per_100_residents_^-1`,
data = LonBoroughstrans,
tol.solve=4.39317e-27,
nb2listw(LB_nb, style="C"),
method = "eigen")
tidy(slag_dv_model3_queen)
## # A tibble: 6 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 rho -9.60e- 3 5.03e-2 -0.191 8.49e- 1
## 2 (Intercept) -4.03e- 2 8.49e-2 -0.475 6.35e- 1
## 3 `Median_annual_income _^-2.5` 1.05e+10 1.63e+9 6.44 1.21e-10
## 4 log_Population_per_square_km 6.02e- 2 1.77e-2 3.40 6.83e- 4
## 5 Percentage_of_homes_with_deficiency_in~ 7.64e- 2 2.80e-2 2.73 6.35e- 3
## 6 `Fast_food_outlets_per_100_residents_^~ -4.88e- 3 2.04e-3 -2.39 1.69e- 2
glance(slag_dv_model3_queen)
## # A tibble: 1 x 6
## r.squared AIC BIC deviance logLik nobs
## <dbl> <dbl> <dbl> <dbl> <dbl> <int>
## 1 0.683 -148. -137. 0.0144 80.8 33
summary(slag_dv_model3_queen)
##
## Call:lagsarlm(formula = `Percentage_of _Obese_Children_in_Year_6` ~
## `Median_annual_income _^-2.5` + log_Population_per_square_km +
## Percentage_of_homes_with_deficiency_in_access_to_nature +
## `Fast_food_outlets_per_100_residents_^-1`, data = LonBoroughstrans,
## listw = nb2listw(LB_nb, style = "C"), method = "eigen", tol.solve = 4.39317e-27)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.05087565 -0.01279994 -0.00012501 0.01126475 0.05046705
##
## Type: lag
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error
## (Intercept) -4.0342e-02 8.4900e-02
## `Median_annual_income _^-2.5` 1.0466e+10 1.6255e+09
## log_Population_per_square_km 6.0230e-02 1.7734e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 7.6398e-02 2.7995e-02
## `Fast_food_outlets_per_100_residents_^-1` -4.8814e-03 2.0441e-03
## z value Pr(>|z|)
## (Intercept) -0.4752 0.6346645
## `Median_annual_income _^-2.5` 6.4386 1.206e-10
## log_Population_per_square_km 3.3964 0.0006829
## Percentage_of_homes_with_deficiency_in_access_to_nature 2.7290 0.0063530
## `Fast_food_outlets_per_100_residents_^-1` -2.3881 0.0169355
##
## Rho: -0.0095987, LR test value: 0.035269, p-value: 0.85103
## Asymptotic standard error: 0.050333
## z-value: -0.1907, p-value: 0.84876
## Wald statistic: 0.036368, p-value: 0.84876
##
## Log likelihood: 80.7873 for lag model
## ML residual variance (sigma squared): 0.00043765, (sigma: 0.02092)
## Number of observations: 33
## Number of parameters estimated: 7
## AIC: -147.57, (AIC for lm: -149.54)
## LM test for residual autocorrelation
## test value: 9.8723e-07, p-value: 0.99921
## Spatial lag model for knn-4
slag_dv_model3_knn4 <- lagsarlm(`Percentage_of _Obese_Children_in_Year_6` ~ `Median_annual_income _^-2.5` +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
`Fast_food_outlets_per_100_residents_^-1`,
data = LonBoroughstrans,
tol.solve = 2.46661e-27,
nb2listw(LB_knn, style="C"),
method = "eigen")
tidy(slag_dv_model3_knn4)
## # A tibble: 6 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 rho 2.70e-1 1.43e-1 1.89 5.86e-2
## 2 (Intercept) -8.73e-2 8.55e-2 -1.02 3.07e-1
## 3 `Median_annual_income _^-2.5` 9.61e+9 1.60e+9 6.02 1.75e-9
## 4 log_Population_per_square_km 5.66e-2 1.64e-2 3.45 5.54e-4
## 5 Percentage_of_homes_with_deficiency_i~ 6.66e-2 2.65e-2 2.51 1.19e-2
## 6 `Fast_food_outlets_per_100_residents_~ -4.46e-3 1.94e-3 -2.30 2.13e-2
glance(slag_dv_model3_knn4)
## # A tibble: 1 x 6
## r.squared AIC BIC deviance logLik nobs
## <dbl> <dbl> <dbl> <dbl> <dbl> <int>
## 1 0.716 -151. -140. 0.0129 82.4 33
summary(slag_dv_model3_knn4)
##
## Call:lagsarlm(formula = `Percentage_of _Obese_Children_in_Year_6` ~
## `Median_annual_income _^-2.5` + log_Population_per_square_km +
## Percentage_of_homes_with_deficiency_in_access_to_nature +
## `Fast_food_outlets_per_100_residents_^-1`, data = LonBoroughstrans,
## listw = nb2listw(LB_knn, style = "C"), method = "eigen",
## tol.solve = 2.46661e-27)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0387843 -0.0128332 -0.0026774 0.0155328 0.0383323
##
## Type: lag
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error
## (Intercept) -8.7341e-02 8.5472e-02
## `Median_annual_income _^-2.5` 9.6069e+09 1.5961e+09
## log_Population_per_square_km 5.6630e-02 1.6399e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 6.6642e-02 2.6503e-02
## `Fast_food_outlets_per_100_residents_^-1` -4.4645e-03 1.9383e-03
## z value Pr(>|z|)
## (Intercept) -1.0219 0.3068455
## `Median_annual_income _^-2.5` 6.0190 1.755e-09
## log_Population_per_square_km 3.4533 0.0005538
## Percentage_of_homes_with_deficiency_in_access_to_nature 2.5145 0.0119206
## `Fast_food_outlets_per_100_residents_^-1` -2.3033 0.0212636
##
## Rho: 0.26967, LR test value: 3.1913, p-value: 0.074032
## Asymptotic standard error: 0.1426
## z-value: 1.8911, p-value: 0.058606
## Wald statistic: 3.5764, p-value: 0.058606
##
## Log likelihood: 82.36531 for lag model
## ML residual variance (sigma squared): 0.00039205, (sigma: 0.0198)
## Number of observations: 33
## Number of parameters estimated: 7
## AIC: -150.73, (AIC for lm: -149.54)
## LM test for residual autocorrelation
## test value: 3.8276, p-value: 0.050415
We’ll check the residuals:
# Residuals from spatial lag model
LonBoroughstrans <- LonBoroughstrans %>%
mutate(slag_dv_model3_knn_resids = residuals(slag_dv_model3_knn4))
KNN4Moran <- LonBoroughstrans %>%
st_drop_geometry()%>%
dplyr::select(slag_dv_model3_knn_resids)%>%
pull()%>%
moran.test(., LB.knn_4_weight)%>%
tidy()
KNN4Moran
## # A tibble: 1 x 7
## estimate1 estimate2 estimate3 statistic p.value method alternative
## <dbl> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 -0.177 -0.0312 0.0113 -1.37 0.915 Moran I test unde~ greater
LonBoroughstrans <- LonBoroughstrans %>%
mutate(slag_dv_model3_queen_resids = residuals(slag_dv_model3_queen))
QueenMoran <- LonBoroughstrans %>%
st_drop_geometry()%>%
dplyr::select(slag_dv_model3_queen_resids)%>%
pull()%>%
moran.test(., LB.queens_weight)%>%
tidy()
QueenMoran
## # A tibble: 1 x 7
## estimate1 estimate2 estimate3 statistic p.value method alternative
## <dbl> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
## 1 0.000118 -0.0312 0.0123 0.283 0.389 Moran I test unde~ greater
Then, we’ll run Spatial Error Model:
# SEM for Queen
SEM_model3_queen <- errorsarlm(`Percentage_of _Obese_Children_in_Year_6` ~ `Median_annual_income _^-2.5` +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
`Fast_food_outlets_per_100_residents_^-1`,
data = LonBoroughstrans,
tol.solve = 3.19139e-27,
nb2listw(LB_nb, style="C"),
method = "eigen")
tidy(SEM_model3_queen)
## # A tibble: 6 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -3.93e- 2 8.48e-2 -0.463 6.43e- 1
## 2 `Median_annual_income _^-2.5` 1.04e+10 1.62e+9 6.44 1.22e-10
## 3 log_Population_per_square_km 5.95e- 2 1.73e-2 3.43 6.01e- 4
## 4 Percentage_of_homes_with_deficiency_in~ 7.55e- 2 2.77e-2 2.73 6.41e- 3
## 5 `Fast_food_outlets_per_100_residents_^~ -4.88e- 3 2.04e-3 -2.38 1.71e- 2
## 6 lambda -2.17e- 3 2.50e-1 -0.00866 9.93e- 1
glance(SEM_model3_queen)
## # A tibble: 1 x 6
## r.squared AIC BIC deviance logLik nobs
## <dbl> <dbl> <dbl> <dbl> <dbl> <int>
## 1 0.682 -148. -137. 0.0145 80.8 33
summary(SEM_model3_queen)
##
## Call:errorsarlm(formula = `Percentage_of _Obese_Children_in_Year_6` ~
## `Median_annual_income _^-2.5` + log_Population_per_square_km +
## Percentage_of_homes_with_deficiency_in_access_to_nature +
## `Fast_food_outlets_per_100_residents_^-1`, data = LonBoroughstrans,
## listw = nb2listw(LB_nb, style = "C"), method = "eigen", tol.solve = 3.19139e-27)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0505145 -0.0127568 0.0006266 0.0115140 0.0500627
##
## Type: error
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error
## (Intercept) -3.9251e-02 8.4785e-02
## `Median_annual_income _^-2.5` 1.0443e+10 1.6223e+09
## log_Population_per_square_km 5.9462e-02 1.7329e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 7.5513e-02 2.7699e-02
## `Fast_food_outlets_per_100_residents_^-1` -4.8759e-03 2.0448e-03
## z value Pr(>|z|)
## (Intercept) -0.4630 0.6433995
## `Median_annual_income _^-2.5` 6.4370 1.219e-10
## log_Population_per_square_km 3.4313 0.0006006
## Percentage_of_homes_with_deficiency_in_access_to_nature 2.7262 0.0064066
## `Fast_food_outlets_per_100_residents_^-1` -2.3845 0.0171024
##
## Lambda: -0.0021665, LR test value: 5.1147e-05, p-value: 0.99429
## Asymptotic standard error: 0.25009
## z-value: -0.0086627, p-value: 0.99309
## Wald statistic: 7.5043e-05, p-value: 0.99309
##
## Log likelihood: 80.7697 for error model
## ML residual variance (sigma squared): 0.00043812, (sigma: 0.020931)
## Number of observations: 33
## Number of parameters estimated: 7
## AIC: -147.54, (AIC for lm: -149.54)
## SEM for KNN4
SEM_model3_knn4 <- errorsarlm(`Percentage_of _Obese_Children_in_Year_6` ~ `Median_annual_income _^-2.5` +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
`Fast_food_outlets_per_100_residents_^-1`,
data = LonBoroughstrans,
tol.solve = 3.19139e-27,
nb2listw(LB_knn, style="C"),
method = "eigen")
tidy(SEM_model3_knn4)
## # A tibble: 6 x 5
## term estimate std.error statistic p.value
## <chr> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) -6.94e- 2 8.21e-2 -0.845 3.98e- 1
## 2 `Median_annual_income _^-2.5` 1.11e+10 1.44e+9 7.69 1.53e-14
## 3 log_Population_per_square_km 6.62e- 2 1.65e-2 4.00 6.25e- 5
## 4 Percentage_of_homes_with_deficiency_in~ 8.50e- 2 2.72e-2 3.13 1.76e- 3
## 5 `Fast_food_outlets_per_100_residents_^~ -5.12e- 3 2.03e-3 -2.52 1.16e- 2
## 6 lambda -2.63e- 1 2.82e-1 -0.935 3.50e- 1
glance(SEM_model3_knn4)
## # A tibble: 1 x 6
## r.squared AIC BIC deviance logLik nobs
## <dbl> <dbl> <dbl> <dbl> <dbl> <int>
## 1 0.692 -148. -138. 0.0141 81.0 33
summary(SEM_model3_knn4)
##
## Call:errorsarlm(formula = `Percentage_of _Obese_Children_in_Year_6` ~
## `Median_annual_income _^-2.5` + log_Population_per_square_km +
## Percentage_of_homes_with_deficiency_in_access_to_nature +
## `Fast_food_outlets_per_100_residents_^-1`, data = LonBoroughstrans,
## listw = nb2listw(LB_knn, style = "C"), method = "eigen",
## tol.solve = 3.19139e-27)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0510293 -0.0099604 -0.0024567 0.0091206 0.0501923
##
## Type: error
## Coefficients: (asymptotic standard errors)
## Estimate Std. Error
## (Intercept) -6.9351e-02 8.2093e-02
## `Median_annual_income _^-2.5` 1.1097e+10 1.4439e+09
## log_Population_per_square_km 6.6198e-02 1.6536e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 8.5043e-02 2.7185e-02
## `Fast_food_outlets_per_100_residents_^-1` -5.1189e-03 2.0286e-03
## z value Pr(>|z|)
## (Intercept) -0.8448 0.398227
## `Median_annual_income _^-2.5` 7.6851 1.532e-14
## log_Population_per_square_km 4.0033 6.247e-05
## Percentage_of_homes_with_deficiency_in_access_to_nature 3.1284 0.001758
## `Fast_food_outlets_per_100_residents_^-1` -2.5234 0.011623
##
## Lambda: -0.26334, LR test value: 0.44141, p-value: 0.50644
## Asymptotic standard error: 0.28168
## z-value: -0.9349, p-value: 0.34984
## Wald statistic: 0.87403, p-value: 0.34984
##
## Log likelihood: 80.99038 for error model
## ML residual variance (sigma squared): 0.00042717, (sigma: 0.020668)
## Number of observations: 33
## Number of parameters estimated: 7
## AIC: -147.98, (AIC for lm: -149.54)
4. GWR model
# Set projections
st_crs(LonBoroughstrans) = 27700
## Warning: st_crs<- : replacing crs does not reproject data; use st_transform for
## that
LonBoroughsSPtrans <- LonBoroughstrans %>%
as(., "Spatial")
## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj
## = prefer_proj): Discarded datum Unknown based on Airy 1830 ellipsoid in CRS
## definition
## Warning in showSRID(SRS_string, format = "PROJ", multiline = "NO", prefer_proj =
## prefer_proj): Discarded datum OSGB 1936 in CRS definition
st_crs(coordsB) = 27700
## Warning: st_crs<- : replacing crs does not reproject data; use st_transform for
## that
coordsBSP <- coordsB %>%
as(., "Spatial")
## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj
## = prefer_proj): Discarded datum Unknown based on Airy 1830 ellipsoid in CRS
## definition
## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj =
## prefer_proj): Discarded datum OSGB 1936 in CRS definition
coordsBSP
## class : SpatialPoints
## features : 33
## extent : 507889.7, 554049, 163541.2, 196420.8 (xmin, xmax, ymin, ymax)
## crs : +proj=tmerc +lat_0=49 +lon_0=-2 +k=0.9996012717 +x_0=400000 +y_0=-100000 +ellps=airy +units=m +no_defs
# calculate bandwitch
GWRbandwidthtrans <- gwr.sel(Percentage_of._Obese_Children_in_Year_6 ~ Median_annual_income._..2.5 +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
Fast_food_outlets_per_100_residents_..1,
data = LonBoroughsSPtrans,
coords=coordsBSP,
adapt=T)
## Warning in gwr.sel(Percentage_of._Obese_Children_in_Year_6 ~
## Median_annual_income._..2.5 + : data is Spatial* object, ignoring coords
## argument
## Adaptive q: 0.381966 CV score: 0.03343324
## Adaptive q: 0.618034 CV score: 0.03528038
## Adaptive q: 0.236068 CV score: 0.03094752
## Adaptive q: 0.145898 CV score: 0.03079962
## Adaptive q: 0.1784092 CV score: 0.02984946
## Adaptive q: 0.189451 CV score: 0.02991537
## Adaptive q: 0.1802365 CV score: 0.02982284
## Adaptive q: 0.1825907 CV score: 0.02980696
## Adaptive q: 0.183218 CV score: 0.02981068
## Adaptive q: 0.1822075 CV score: 0.02980534
## Adaptive q: 0.1814547 CV score: 0.02980813
## Adaptive q: 0.1820962 CV score: 0.02980496
## Adaptive q: 0.1818512 CV score: 0.02980428
## Adaptive q: 0.1816997 CV score: 0.02980546
## Adaptive q: 0.1819448 CV score: 0.02980452
## Adaptive q: 0.1817933 CV score: 0.02980447
## Adaptive q: 0.1818919 CV score: 0.02980438
## Adaptive q: 0.1818512 CV score: 0.02980428
GWRbandwidthtrans
## [1] 0.1818512
# run the GWR model
gwr.modeltrans = gwr(Percentage_of._Obese_Children_in_Year_6 ~ Median_annual_income._..2.5 +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
Fast_food_outlets_per_100_residents_..1,
data = LonBoroughsSPtrans,
coords=coordsBSP,
adapt=GWRbandwidthtrans,
hatmatrix = TRUE,
se.fit = TRUE)
## Warning in gwr(Percentage_of._Obese_Children_in_Year_6 ~
## Median_annual_income._..2.5 + : data is Spatial* object, ignoring coords
## argument
## Warning in proj4string(data): CRS object has comment, which is lost in output
## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj
## = prefer_proj): Discarded datum Unknown based on Airy 1830 ellipsoid in CRS
## definition
Let’s print the results of the GWR model:
gwr.modeltrans
## Call:
## gwr(formula = Percentage_of._Obese_Children_in_Year_6 ~ Median_annual_income._..2.5 +
## log_Population_per_square_km + Percentage_of_homes_with_deficiency_in_access_to_nature +
## Fast_food_outlets_per_100_residents_..1, data = LonBoroughsSPtrans,
## coords = coordsBSP, adapt = GWRbandwidthtrans, hatmatrix = TRUE,
## se.fit = TRUE)
## Kernel function: gwr.Gauss
## Adaptive quantile: 0.1818512 (about 6 of 33 data points)
## Summary of GWR coefficient estimates at data points:
## Min. 1st Qu.
## X.Intercept. -1.1758e-01 -5.8218e-02
## Median_annual_income._..2.5 5.8826e+09 7.1845e+09
## log_Population_per_square_km 2.6732e-02 4.3915e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 4.8259e-02 6.3068e-02
## Fast_food_outlets_per_100_residents_..1 -6.8343e-03 -6.2540e-03
## Median 3rd Qu.
## X.Intercept. -1.8931e-02 1.4746e-02
## Median_annual_income._..2.5 8.5226e+09 9.4882e+09
## log_Population_per_square_km 5.5738e-02 6.6814e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 7.6677e-02 8.8804e-02
## Fast_food_outlets_per_100_residents_..1 -3.2608e-03 -1.4918e-03
## Max. Global
## X.Intercept. 8.4562e-02 -3.910e-02
## Median_annual_income._..2.5 1.1447e+10 1.044e+10
## log_Population_per_square_km 8.3678e-02 5.940e-02
## Percentage_of_homes_with_deficiency_in_access_to_nature 9.8878e-02 7.540e-02
## Fast_food_outlets_per_100_residents_..1 6.6187e-04 -4.900e-03
## Number of data points: 33
## Effective number of parameters (residual: 2traceS - traceS'S): 14.50372
## Effective degrees of freedom (residual: 2traceS - traceS'S): 18.49628
## Sigma (residual: 2traceS - traceS'S): 0.02021397
## Effective number of parameters (model: traceS): 11.35138
## Effective degrees of freedom (model: traceS): 21.64862
## Sigma (model: traceS): 0.01868438
## Sigma (ML): 0.01513341
## AICc (GWR p. 61, eq 2.33; p. 96, eq. 4.21): -141.4577
## AIC (GWR p. 96, eq. 4.22): -171.5948
## Residual sum of squares: 0.007557667
## Quasi-global R2: 0.8340129
resultstrans <- as.data.frame(gwr.modeltrans$SDF)
names(resultstrans)
## [1] "sum.w"
## [2] "X.Intercept."
## [3] "Median_annual_income._..2.5"
## [4] "log_Population_per_square_km"
## [5] "Percentage_of_homes_with_deficiency_in_access_to_nature"
## [6] "Fast_food_outlets_per_100_residents_..1"
## [7] "X.Intercept._se"
## [8] "Median_annual_income._..2.5_se"
## [9] "log_Population_per_square_km_se"
## [10] "Percentage_of_homes_with_deficiency_in_access_to_nature_se"
## [11] "Fast_food_outlets_per_100_residents_..1_se"
## [12] "gwr.e"
## [13] "pred"
## [14] "pred.se"
## [15] "localR2"
## [16] "X.Intercept._se_EDF"
## [17] "Median_annual_income._..2.5_se_EDF"
## [18] "log_Population_per_square_km_se_EDF"
## [19] "Percentage_of_homes_with_deficiency_in_access_to_nature_se_EDF"
## [20] "Fast_food_outlets_per_100_residents_..1_se_EDF"
## [21] "pred.se.1"
Moran’s I for GWR:
gwr.morantest(gwr.modeltrans, LB.queens_weight)
## Warning in gwr.morantest(gwr.modeltrans, LB.queens_weight): LB.queens_weightnot
## row standardised
##
## Leung et al. 2000 three moment approximation for Moran's I
##
## data: GWR residuals
## statistic = 1106.5, df = 1130.9, p-value = 0.3074
## sample estimates:
## I
## -0.09815888
gwr.morantest(gwr.modeltrans, LB.knn_4_weight)
## Warning in gwr.morantest(gwr.modeltrans, LB.knn_4_weight): LB.knn_4_weightnot
## row standardised
##
## Leung et al. 2000 three moment approximation for Moran's I
##
## data: GWR residuals
## statistic = 4305.5, df = 4329.7, p-value = 0.3999
## sample estimates:
## I
## -0.1235569
Monte Carlo approach:
res.mont <- gwr.montecarlo(Percentage_of._Obese_Children_in_Year_6 ~ Median_annual_income._..2.5 +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
Fast_food_outlets_per_100_residents_..1,
data = LonBoroughsSPtrans, bw = 30, adaptive = TRUE,
nsims = 95, kernel="gaussian")
##
## Tests based on the Monte Carlo significance test
##
## p-value
## (Intercept) 0.5521
## Median_annual_income._..2.5 0.5938
## log_Population_per_square_km 0.4688
## Percentage_of_homes_with_deficiency_in_access_to_nature 0.5833
## Fast_food_outlets_per_100_residents_..1 0.1042
bw <- bw.gwr(Percentage_of._Obese_Children_in_Year_6 ~ Median_annual_income._..2.5 +
log_Population_per_square_km +
Percentage_of_homes_with_deficiency_in_access_to_nature +
Fast_food_outlets_per_100_residents_..1,
data = LonBoroughsSPtrans,
approach = "AIC", kernel="gaussian")
## Fixed bandwidth: 28625 AICc value: -146.554
## Fixed bandwidth: 17694.76 AICc value: -146.4194
## Fixed bandwidth: 35380.26 AICc value: -146.5028
## Fixed bandwidth: 24450.02 AICc value: -146.5747
## Fixed bandwidth: 21869.74 AICc value: -146.5653
## Fixed bandwidth: 26044.72 AICc value: -146.5697
## Fixed bandwidth: 23464.44 AICc value: -146.5745
## Fixed bandwidth: 25059.14 AICc value: -146.5735
## Fixed bandwidth: 24073.56 AICc value: -146.575
## Fixed bandwidth: 23840.9 AICc value: -146.5749
res.mont
## p-value
## (Intercept) 0.5520833
## Median_annual_income._..2.5 0.5937500
## log_Population_per_square_km 0.4687500
## Percentage_of_homes_with_deficiency_in_access_to_nature 0.5833333
## Fast_food_outlets_per_100_residents_..1 0.1041667
bw
## [1] 24073.56
Attach coefficients to original SF and plot them:
LonBoroughs2 <- LonBoroughstrans %>%
mutate(coefMedinc = resultstrans$Median_annual_income._..2.5,
coefPopdens = resultstrans$log_Population_per_square_km ,
coefaccnat = resultstrans$Percentage_of_homes_with_deficiency_in_access_to_nature,
coeffastfood = resultstrans$Fast_food_outlets_per_100_residents_..1)
tmap_mode("view")
## tmap mode set to interactive viewing
tm_shape(LonBoroughs2) +
tm_polygons(col = "coefMedinc", palette = "RdBu", alpha = 0.6)
tm_shape(LonBoroughs2) +
tm_polygons(col = "coefPopdens", palette = "RdBu", alpha = 0.6)
tm_shape(LonBoroughs2) +
tm_polygons(col = "coefaccnat", palette = "RdBu", alpha = 0.6)
tm_shape(LonBoroughs2) +
tm_polygons(col = "coeffastfood", palette = "RdBu", alpha = 0.6, midpoint = NA)
Run the significance test:
sigTest1 = abs(gwr.modeltrans$SDF$Median_annual_income._..2.5)-2 * gwr.modeltrans$SDF$Median_annual_income._..2.5_se
sigTest2 = abs(gwr.modeltrans$SDF$log_Population_per_square_km)-2 * gwr.modeltrans$SDF$log_Population_per_square_km_se
sigTest3 = abs(gwr.modeltrans$SDF$Percentage_of_homes_with_deficiency_in_access_to_nature)-2 * gwr.modeltrans$SDF$Percentage_of_homes_with_deficiency_in_access_to_nature_se
sigTest4 = abs(gwr.modeltrans$SDF$Fast_food_outlets_per_100_residents_..1)-2 * gwr.modeltrans$SDF$Fast_food_outlets_per_100_residents_..1_se
Finally, let’s plot significance results:
tmap_mode("view")
## tmap mode set to interactive viewing
LonBoroughs2 <- LonBoroughs2 %>%
mutate(GWRMedincSig = sigTest1)
tm_shape(LonBoroughs2) +
tm_polygons(col = "GWRMedincSig", palette = "RdYlBu", alpha = 0.6)
LonBoroughs2 <- LonBoroughs2 %>%
mutate(GWRPopdensSig = sigTest2)
tm_shape(LonBoroughs2) +
tm_polygons(col = "GWRPopdensSig", palette = "RdYlBu", alpha = 0.6)
## Variable(s) "GWRPopdensSig" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.
